On $\mathbb{P}^2$, we have two full strong exceptional collections:
$\{\mathcal{O}_{\mathbb{P}^2},\Omega_{\mathbb{P}^2}(2),\mathcal{O}_{\mathbb{P}^2}(1)\}$ and $\{\mathcal{O}_{\mathbb{P}^2}(-2), \mathcal{O}_{\mathbb{P}^2}(-1), \mathcal{O}_{\mathbb{P}^2}\}$.
We use $B_1,B_2$ to denote the endomorphism algebra of the direct sum of the two collections respectively. Consider a quiver $Q$ with three vertices $v_0,v_1,v_2$ and three arrows $a_i$ from $v_0$ to $v_1$ and $b_i$ from $v_1$ to $v_2$. Define two ideals $J_1$,$J_2$ where $J_1$ is generated by $b_ia_j+b_ja_i$ while $J_2$ is generated by $b_ia_j-b_ja_i$. Then $B_1\simeq \mathbb{C}Q/J_1$ and $B_2\simeq \mathbb{C}Q/J_2$.
In the paper of [Ryo Ohkawa][1]
[1]: https://www.jstage.jst.go.jp/article/kodaimath/33/2/33_2_329/_article/-char/ja/ he computed that using the equivalence $RHom(\mathcal{O}_{\mathbb{P}^2}\oplus\Omega_{\mathbb{P}^2}(2)\oplus\mathcal{O}_{\mathbb{P}^2}(1),-):D^b(P^2)\to D^b(mod-B_1)$, the preimage of the abelian category $(mod-B_1)$ is the extension closure
$\mathcal{A}=<\mathcal{O}_{\mathbb{P}^2}(-2)[2], \mathcal{O}_{\mathbb{P}^2}(-1)[1], \mathcal{O}_{\mathbb{P}^2}>$
Objects in this category will be complexes $\mathcal{O}_{\mathbb{P}^2}(-2)^{\oplus a}\to\mathcal{O}_{\mathbb{P}^2}(-1)^{\oplus b}\to\mathcal{O}_{\mathbb{P}^2}^{\oplus c}$ which give representations of $Q$ with relation $J_2$. So I believe $\mathcal{A}\simeq (B_2-mod)$. (EDIT: This is wrong. I did not account for the fact that d^2=0. When this is noted, we obtain $B_1-mod$!)
Noting the symmetry of elements in $J_1$, we see $B_1\simeq B_1^{op}$, thus combining all this it seems we have $(B_2-mod)\simeq (B_1-mod)$, which is a little surprising to me.
My question: Is this an example of Morita equivalence or I made a mistake somewhere? If this is an example, is there a general theory of such a construction using exceptional collections?
Thanks for the help.
